## Journal of Commutative Algebra

### Cofiniteness and non-vanishing of local cohomology modules

#### Abstract

Let $R$ be a commutative Noetherian local ring, $I$ an ideal of $R$, and let $M$ be a non-zero finitely generated $R$-module. In this paper, we establish some new properties of the local cohomology modules $H^i_I(M)$, $i\geq 0$. In particular, we show that if $(R,\mathfrak{m})$ is a Noetherian local integral domain of dimension $d \leq 4$ which is a homomorphic image of a Cohen-Macaulay ring and $x_1,\ldots,x_n$ is a part of a system of parameters for $R$, then for all $i\geq0$, the $R$-modules $H^i_{I}(R)$ are $I$-cofinite, where $I=(x_1,\ldots,x_n)$. Also, we prove that if $(R,\mathfrak{m})$ is a Noetherian local ring of dimension~$d$ and $x_1,\ldots,x_t$ is a part of a system of parameters for $R$, then $H^{d-t}_{\mathfrak{m}}(H^t_{(x_1,\ldots,x_t)}(R))\neq 0$. In particular, $\mu^{d-t}(\mathfrak{m},H^t_{(x_1,\ldots,x_t)}(R))\neq 0$ and ${\rm injdim}_R(H^t_{(x_1,\ldots,x_t)}(R))\geq d-t$.

#### Article information

Source
J. Commut. Algebra Volume 6, Number 3 (2014), 305-321.

Dates
First available in Project Euclid: 17 November 2014

https://projecteuclid.org/euclid.jca/1416233320

Digital Object Identifier
doi:10.1216/JCA-2014-6-3-305

Mathematical Reviews number (MathSciNet)
MR3278806

Zentralblatt MATH identifier
1299.13019

#### Citation

Bagheriyeh, Iraj; Bahmanpour, Kamal; A'Zami, Jafar. Cofiniteness and non-vanishing of local cohomology modules. J. Commut. Algebra 6 (2014), no. 3, 305--321. doi:10.1216/JCA-2014-6-3-305. https://projecteuclid.org/euclid.jca/1416233320.

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